謝慕雲 asked in 科學數學 · 1 decade ago

請幫我解這題高微題目

Let R be a regular region in R3 with piecewise smooth boundary.

Show that the volume of R is 1/3 ∫∫ R F*n dA where F( x , y , z ) = Xi + Yj + Zk.

我已經知道答案為3A,但非常需要過程,謝謝

Update:

修正一下,是 ∫ ∫ R(下標) F*n dA

Update 2:

抱歉,我搞錯了,這題是證明題,答案不是3A

1 Answer

Rating
  • 天助
    Lv 7
    1 decade ago
    Favorite Answer

    F=(x,y,z), divF=1+1+1=3

    Let S be the boundary of the region R(volume of R is V),then

    ∫∫_S F‧n dA=∫∫∫_R divF dv

    =∫∫∫_R 3dv=3∫∫∫_R dv=3V

    so, V=(1/3)∫∫_S F‧n dA

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